Thanks to [1]} , we infer that \(\mathcal {A}_\kappa ^\sigma +\mathcal {B}\)  is a maximal monotone operator with domain
\(D(\mathcal {A}_\kappa ^\sigma +\mathcal {B})=D(\mathcal {A}_\kappa ^\sigma )\) , which coincides with \( \partial \Phi ^\star \) . In light of our assumptions, we recall that
\((\phi ,\psi ) \in L^2(0,T; \mathcal {D}_\sigma )\cap H^1(0,T;\mathcal {D}_\sigma ^{\prime })\) . Then, since \(\phi |_\Gamma = \sigma \psi \)  on \(\Gamma \) , we observe that
\(\partial \Phi ^\star (\phi ,\psi )=(-\Delta \phi +\phi , -\kappa \Delta _\Gamma \psi + \sigma \partial _\mathbf {n}\phi +\psi )\in L^2(0,T;\mathcal {D}_\sigma ).\) 
